Integration using Partial Fractions

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JEE Advanced 1999

Integrate $\int \frac{x ^{3} + 3 x + 2}{( x ^{2} + 1 ) ^{2} ( x + 1 )} d x$

JEE Main 2021 (31 Aug Shift 2)

If $\int \frac{s i n x}{s i n ^{3} x + c o s ^{3} x} d x = α lo g_{e} ∣1 + tan x ∣ + β lo g_{e} ∣1 - tan x + tan^{2} x ∣ + γ tan^{- 1} (\frac{2 t a n x - 1}{3}) + C$ , when $C$ is constant of integration, then the value of $18 (α + β + γ^{2})$ is.

JEE Main 2021 (20 July Shift 2)

For $k \in N$ , let $\frac{1}{α ( α + 1 ) ( α + 2 ) \dots ( α + 20 )} = \sum_{k = 0}^{20} \frac{A _{k}}{α + k}$ , where $α > 0$ . Then the value of $100 (\frac{A _{14} + A _{15}}{A _{13}})^{2}$ is equal to